bajaj562chpt7

bajaj562chpt7 - CHAPTER 7 Basic Concepts and Kinematics of...

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1 CHAPTER 7 Basic Concepts and Kinematics of Rigid Body Motion 7.1 Degrees-of-freedom : (of a rigid body) Consider three unconstrained particles Y Z X O r 1 F 1 r C r 3 F 2 F 3 m 3 m 2 m 1 m C f 12 f 21 f 23 f 32 3
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2 The positions are defined by i = 1, 2, 3. degrees of freedom = n = 9 = 3N Now, constrain the particles (three particles placed at the corners of a triangle whose sides are formed by rigid massless rods) k z j y i x r i i i i Y Z X O r 1 r 3 m 3 m 2 m 1 l 1 l 2 l 3
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3 Now: there are three constraints n = 3(N) - 3 = 6 degrees of freedom . Now: as another particle is introduced, its position is specified by 3 additional coordinates, but also have 3 additional constraints. four rigidly connected particles also have only 6 degrees of freedom . 1 2 1 3 2 2 1 3 3 2 2 2 2 2 1 2 1 2 1 1 ;; or ( ) ( ) ( ) 0, . r r l r r l r r l x x y y z z l etc
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4 In general: a rigid body (any collection of particles whose relative positions are fixed) has 6 degrees of freedom . translational motion of a point on the body - specified by 3 translational degrees of freedom . rotational motion about the specified point - 3 rotational degrees of freedom .
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5 Laws of motion for a system of particles ( extended to a rigid body ) a) translational motion of the C.M. (3 degrees of freedom) b) about C.M. or an inertially fixed point rotational motion about the C.M. (3 rotational degrees of freedom) c F mr d MH dt
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6 7.2 Moments of Inertia Recall : the notation and definitions The equation for moment about an arbitrary point P is: p c p p r m H dt d M Z X Y O r P r C r i F i m i m C f i2 i C P
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7 Now, the angular momentum about the point P is where - velocity of as viewed by a non-rotating observer translating with P. ( relative velocity in an inertial frame ) For a rigid body - suppose that P is fixed in the body = constant, and where - angular velocity of the body. i i i N 1 i p m H i i m i i i
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8 Thus, the angular momentum about P is By analogy: for a rigid body rotating with angular velocity   i i N 1 t i p m H dV ) r ( dm ; ) ( ) ( H V r   Z X Y O r P r dm P V
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9 We now consider the various cases: Reference point P is at origin : Z X Y O=P dm V 22 Then ( ) [( ) ][ ( ) ] [ ( ) ] x y z x y z x yz x y z xi yj zk i j k xy xz i xy x z yz j xz yz x y k     
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10 Let us define : These are the moments of inertia P O point reference the through axis z about I y I x I zz yy xx )dV z y ( I 2 2 V )dV z x ( I ; )dV z x ( I 2 V 2 2 V 2
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11 Similarly, we define products of Inertia : ( angular momentum vector for the body , or angular momentum about P) V yx xy dV I I V zx xz I I V zy yz I I k H j H i H H z y x p
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12 Here, In compact notation: where
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bajaj562chpt7 - CHAPTER 7 Basic Concepts and Kinematics of...

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